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Smith (1969) Annular-Mist Slip Ratio & Void Fraction Formula: Entrained Liquid e_k = 0.40
Calculate steam-water and refrigerant boiling cross-sectional void fraction alpha using S. L. Smith’s (1969) Momentum-Equalized Annular-Mist Slip Ratio S_Smith with entrained droplet fraction e_k = 0.40.
Read the guide →Healthcare & Clinical CalculationsSmith-Modified Sgarbossa Criteria for LBBB & Paced Occlusion MI (STEMI)
Evaluate acute coronary occlusion in left bundle branch block (LBBB) or right-ventricular paced rhythms using concordant ST shifts (>= 1 mm) and Stephen Smith’s proportional discordant ST/S ratio (<= -0.25).
Read the guide →Fatigue Durability & Mechanical ReliabilitySmith-Watson-Topper (SWT) & Morrow Fatigue Formula
Calculate Peak Cyclic Stress sigma_max = sigma_a + sigma_m, Smith-Watson-Topper Damage Stress SQRT(MAX(0, sigma_max) * epsilon_a * E), Morrow Mean-Stress Equivalent Amplitude, and Crack-Closure Regime.
Read the guide →SMT Electronics Assembly, SAC305 Lead-Free Reflow Profiling & IPC/JEDEC J-STD-020SMT Reflow Oven Belt Speed, Ramp Slope & TAL Formula
Calculate Preheat Zone Residence Time (s), Preheat Thermal Ramp Slope (°C/s <= 3.0 °C/s limit), Peak Zone Time Above 217 °C SAC305 Liquidus TAL (s), and J-STD-020 Profile Audit.
Read the guide →SMT Electronics Assembly, IPC-7525 Solder-Paste Stencil Design & Reflow YieldSMT Stencil Aperture Area Ratio & Paste Volume Formula
Calculate IPC-7525 Aperture Area Ratio AR = (L*W)/(2*(L+W)*T), Width-to-Foil Aspect Ratio W/T, Wet Solder-Paste Brick Volume (nL), and >= 0.66 Paste-Release Gate.
Read the guide →Healthcare & Clinical CalculatorsSNAPPE-II (Score for Neonatal Acute Physiology with Perinatal Extension-II)
Sum first-12-hour NICU physiologic and perinatal SNAPPE-II subscores (0–162) and compute Douglas Richardson et al.’s (2001) neonatal hospital mortality probability.
Read the guide →Clinical Medicine & PhysiologySNAQ Score Calculator: Simplified Nutritional Appetite Questionnaire (Wilson 4-Item)
Screen older adults for anorexia of aging and 6-month involuntary weight loss (>= 5% and >= 10%) using Margaret-Mary G. Wilson et al.’s (2005) 4-item Simplified Nutritional Appetite Questionnaire (SNAQ, 4–20 points; <= 14 risk cutoff).
Read the guide →Engineering & ScientificSnell’s Law Refraction Angle & Total Internal Reflection (TIR) Critical Angle
Calculate optical refraction angle theta_2 and Total Internal Reflection critical angle theta_c = DEGREES(ASIN(n_2 / n_1)) across planar dielectric interfaces with radian-degree conversion and ASIN domain guards.
Read the guide →EngineeringSnell's Law refraction angle & Total Internal Reflection critical angle
Calculate refracted ray angles DEGREES(ASIN((n1/n2)*SIN(RADIANS(theta1)))) and fiber-optic Total Internal Reflection critical angles safely in degrees.
Read the guide →Engineering & ScienceSoave-Redlich-Kwong (SRK) Equation of State: Alpha Cohesion & Pressure
Calculate Giorgio Soave’s acentric cohesion function α(T_r, ω) = [1 + (0.480 + 1.574ω - 0.176ω^2)(1 - SQRT(T / T_c))]^2 and real-gas SRK pressure P (bar) for hydrocarbon and natural-gas processing.
Read the guide →Health & FitnessSOFA Score (Sequential Organ Failure Assessment) & Sepsis-3 Criterion
Calculate the 6-system (0–24 point) ICU SOFA Score across Respiration, Coagulation, Liver, Cardiovascular, CNS, and Renal subscores, and evaluate Sepsis-3 acute Delta-SOFA >= 2 organ dysfunction.
Read the guide →Soil Chemistry & Agricultural HydrologySoil Sodium Adsorption Ratio (SAR) & ESP Formula
Calculate USDA Salinity Lab Sodium Adsorption Ratio SAR = Na / SQRT((Ca + Mg) / 2) in meq/L, Empirical Exchangeable Sodium Percentage (ESP), and Normal / Saline / Sodic / Saline-Sodic soil class.
Read the guide →Top-Down Excavation Shoring, FHWA GEC-7 Grouted Soil-Nail Wall Pullout & SpacingSoil-Nail Wall Grout Pullout Capacity & Spacing Formula
Calculate Ultimate Grout-Soil Bond Pullout Qu (kN), FHWA ASD Allowable Pullout Resistance (FS = 2.0, kN), Tributary Facing Nail Load (kN), and Pullout Factor of Safety.
Read the guide →Power Grid Operations & Renewable EconomicsSolar Curtailment LCOE & Duck-Curve Ramp Formula
Calculate Curtailment-Adjusted Delivered LCOE = LCOE_gross / (1 - c_curt), Nonlinear Curtailment Cost Penalty c_curt / (1 - c_curt), Duck-Belly Midday Net Load, and Sunset Thermal/BESS Ramp Rate (MW/hr).
Read the guide →EngineeringSolar Declination Angle, Noon Sun Altitude & PV Air Mass (AM) formulas
Explore approximate seasonal declination, geometric solar-noon elevation and a simple relative air-mass approximation.
Read the guide →EngineeringSolar PV Cell Fill Factor (FF) & Power Conversion Efficiency (PCE)
Use measured maximum-power voltage and current to calculate PV output watts, fill factor and conversion efficiency, with consistent irradiance, area and measurement conditions.
Read the guide →Renewable Energy & Power Systems EngineeringSolar PV NOCT Thermal Derating & IEC PR Formula
Calculate operating silicon cell temperature T_cell = T_amb + ((NOCT - 20) / 800) * G, thermal-derated DC array output, and IEC 61724 plant Performance Ratio PR.
Read the guide →EngineeringSolar PV panel array & battery bank (Ah) sizing calculator with ROUNDUP
Build an illustrative daily-energy panel-count and nominal battery-capacity worksheet with explicit loss assumptions. This is not a complete electrical design.
Read the guide →EngineeringSolenoid Magnetic Field & Helmholtz Coil Pair Uniform Field formulas
Calculate axial magnetic flux density inside a long solenoid B = mu_0 * (N / L) * I, midpoint uniform field of a Helmholtz coil pair B = (4/5)^(3/2) * mu_0 * N * I / R, and magnetic energy density.
Read the guide →Financial & AccountingSolow-Swan Growth Model Formula: Steady-State Capital k* & Output y* per Effective Worker
Solve Nobel laureate Robert Solow & Trevor Swan’s 1956 neoclassical growth model under Cobb-Douglas production y = k^alpha for balanced-growth steady-state capital k* = (s / (n + g + delta))^(1 / (1 - alpha)) and output y* = (k*)^alpha.
Read the guide →ScienceSolution Ionic Strength (I) & Debye-Huckel Activity Coefficient
Calculate concentration-based ionic strength from a complete ion list, then explore a dilute-water Debye–Hückel limiting-law estimate at 25 °C.
Read the guide →Industrial Inorganic Chemistry, Glass Raw Materials & Soda AshSolvay Soda Ash Process Salt, Limestone & CaCl2 Yield Formula
Calculate Dense Soda Ash Na2CO3 Production (t/d), Quarried Limestone CaCO3 Kiln Feed (t/d), Byproduct Calcium Chloride CaCl2 (t/d), and Unreacted Brine NaCl Slip (t/d).
Read the guide →Math & TrigSolve a quadratic equation (ax^2 + bx + c = 0) with LET and SQRT
Calculate the discriminant b^2 - 4ac, guard complex/linear cases cleanly, and compute both real roots (-b ± SQRT(d)) / (2a).
Read the guide →Math & TrigSolve a system of two linear equations (Cramer's Rule formula)
Solve a1*x + b1*y = c1 and a2*x + b2*y = c2 directly in cells using determinants and guard parallel or redundant lines where det = 0.
Read the guide →Financial ModelingSolve Implied Loan & Annuity Interest Rate from Payment, Principal, and Term
Back out the periodic interest rate and nominal APR of a level-payment loan or equipment lease using quadratic Newton-Raphson iteration and verify zero payment residual with PMT.
Read the guide →Actuarial Science, Insurance Capital & Solvency II RegulationSolvency II BSCR Correlation Diversification & Coverage Formula
Calculate EIOPA Solvency II Standard-Formula Undiversified SCR Sum, Variance-Covariance Correlated Basic Solvency Capital Requirement SQRT(SCR_1^2 + SCR_2^2 + 2*rho*SCR_1*SCR_2), Diversification Benefit, and Solvency Coverage Ratio.
Read the guide →Engineering & ScienceSommerfeld Electronic Heat Capacity (γT) & Phonon Crossover Temperature
Calculate the free-electron Sommerfeld coefficient γ = (π^2 / 2) * Z * R / T_F (mJ/(mol·K^2)), linear electronic heat capacity C_el = γT, and cryogenic electron-phonon crossover temperature T_cross = SQRT(γ / A_ph).
Read the guide →EngineeringSommerfeld Journal-Bearing Number (S) & Petroff Friction Torque
Calculate dimensionless hydrodynamic bearing Sommerfeld Number S = ((r / c)^2) * ((mu * N) / P) for Raimondi-Boyd eccentricity charts and Petroff's concentric viscous friction coefficient & shaft torque.
Read the guide →Lookup & Dynamic ArraysSort a Table by Two Columns (Region Ascending, Revenue Descending) with SORTBY
Order records simultaneously by a primary text category ascending (+1) and a secondary numeric KPI descending (-1) using multi-key SORTBY.
Read the guide →TextSort and deduplicate comma-separated values inside a single cell
Canonicalize messy tag strings ("urgent, billing, api, billing" -> "api, billing, urgent") using TEXTSPLIT, TRANSPOSE, UNIQUE, SORT, and TEXTJOIN.
Read the guide →Date & TimeSort birthdays by month and day (ignoring birth year)
Order team birthdays or annual renewals from January 1 to December 31 regardless of birth year using a numeric MMDD key and SORTBY.
Read the guide →Dynamic ArraysSort by a custom list order (Critical, High, Medium, Low) with SORTBY & XMATCH
Sort tickets, sizes (XS, S, M, L, XL), or pipeline stages in business-defined order instead of alphabetical A–Z order.
Read the guide →Dynamic arraysSORT function: build a live sorted leaderboard without editing source rows
Spill a sorted copy of a multi-column table ordered ascending or descending by any column, updating automatically whenever source numbers change.
Read the guide →Lookup & ReferenceSort SKUs and composite codes by an embedded substring or middle number (SORTBY)
Sort compound identifiers like "WH-1050-EUR" and "ZR-0015-USA" by their middle part number or trailing region suffix using TEXTBEFORE, TEXTAFTER, and SORTBY.
Read the guide →Dynamic ArraysSort text by character length (with alphabetical tie-breaking in SORTBY)
Order product SKUs, ad copy, or domain names from shortest to longest (or longest first) and break equal-length ties A–Z using two-level SORTBY.
Read the guide →Dynamic arraysSORTBY function: sort by multiple columns or by a column excluded from output
Rank data across two or more tie-breaker columns—and return only the columns you want to display—using SORTBY.
Read the guide →Engineering & ScienceSouders-Brown Equation: Knockout-Drum Max Vapor Velocity & Diameter
Size vertical vapor-liquid flash drums, demister pads, and distillation columns against droplet entrainment using V_max = K_SB * SQRT((rho_L - rho_V) / rho_V) (m/s) and minimum vessel inside diameter D_min.
Read the guide →StatisticalSpearman Rank Correlation (rho) formula: outlier-resistant CORREL on ranks
Measure monotonic non-linear relationships and resist extreme outliers by computing Pearson CORREL across RANK columns.
Read the guide →Statistical & PsychometricsSpearman’s Footrule Rank Distance & Normalized Rank Similarity Formula
Measure L1 rank-order disagreement D = SUM(|R_A,i - R_B,i|) and Diaconis-Graham normalized similarity 1 - D / FLOOR(n^2 / 2) across search engine SERPs and judge rankings.
Read the guide →Information Economics & Labor EconomicsSpence Job-Market Signaling Separating Threshold Formula
Compute the single-crossing separating education interval [e_min, e_max], Riley least-cost credential e* = (y_H - y_L)/c_L, and pooling Pareto-dominance cutoff.
Read the guide →Industrial Organization & Supply Chain PricingSpengler Double Marginalization & Channel Fix Formula
Quantify Joseph Spengler’s (1950) vertical double-marginalization profit loss (25%) and consumer-surplus loss (75%) and solve the wholesale w* = c + franchise fee channel fix.
Read the guide →Healthcare & Clinical FormulasSpetzler-Martin Brain AVM Grading Scale (Grades I–V & Spetzler-Ponce Class)
Calculate Robert Spetzler and Neil Martin’s cerebral arteriovenous malformation surgical risk score (1–5: Nidus Size <3 cm = 1, 3–6 cm = 2, >6 cm = 3; Eloquent Cortex = +1; Deep Venous Drainage = +1) and 3-tier Spetzler-Ponce treatment class.
Read the guide →Statistical & PsychometricsSpiegelhalter’s Z-Test for Probability Calibration (Exact Brier Decomposition)
Test whether ML or clinical risk probabilities p_i are calibrated without arbitrary Hosmer-Lemeshow decile binning via David Spiegelhalter’s (1986) exact Z-statistic!
Read the guide →Text & Data CleaningSplit "Width x Height" Dimension Strings into Numeric Columns & Surface Area
Parse messy catalog dimension strings like "24.5 x 18", "120X80 cm", or "60×40" into numeric Width, Height, and Area columns using LOWER, SUBSTITUTE, FIND, LEFT, MID, and VALUE.
Read the guide →FinancialSplit a contract payment or prepaid cost across calendar months (daily proration)
Allocate a hypothetical contract amount proportionally to inclusive active calendar days in each month.
Read the guide →Math & TrigSplit a number into digits and get the digit at any place value
Extract the Nth digit from the left with MID or isolate ones, tens, and hundreds place values numerically with QUOTIENT and MOD.
Read the guide →TextSplit first, middle, and last names (plus "Last, First" comma format)
Extract given names, optional middle names, and surnames cleanly with TEXTBEFORE and TEXTAFTER—and flip "Last, First" payroll strings.
Read the guide →TextSplit L x W x H dimensions ("24 x 18 x 12") into numbers and volume
Parse 2D or 3D carton dimension text into numeric Length, Width, Height, and cubic volume using TEXTBEFORE, TEXTAFTER, and VALUE.
Read the guide →TextSplit numbers from units of measure ("12.5 kg", "2500mg")
Separate an unsigned decimal numeric prefix from a trailing alphabetic unit in controlled measurement strings.
Read the guide →Food, Dairy & Pharmaceutical Particle Engineering, Spray DryingSpray Dryer Water Evaporation, Powder Output & Efficiency Formula
Calculate Bone-Dry Solids Throughput (kg/hr), Commercial Residual-Moisture Powder Rate (kg/hr), Spray-Chamber Water Evaporated (kg/hr), and Ideal Adiabatic Thermal Efficiency (%).
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